Mathematics
The Cauchy-Riemann Equations and Differentiability of Complex Functions
Quick fact
Unlike real functions, a complex function that is differentiable once automatically has derivatives of all orders and can be represented as a power series, because the Cauchy–Riemann equations enforce a rigid structure.
Why this is interesting
You know how to take derivatives of real functions, but what does it mean to differentiate a function that has both a real and an imaginary part? It turns out that this seemingly simple question leads to a surprisingly strict rule.